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We present a theoretical framework of a new problem representation. The quotient structure model is presented for representing different grain-size worlds. Our aim is to explore the relationship between different grain-size worlds and reveal the essence of human hierarchical problem solving skill. The motivation of the research stems from our belief that more human-like characteristics in problem solving should be involved in a formal representation to achieve better performance for computer- based problem solver.
To illustrate the role of the hierarchy we make use of topologic and semi-order space as expository tools and show that attribute-preserving among different grain-size worlds is of central importance in hierarchical problem solving. The complexity of hierarchical problem solving by using the quotient structure model is analyzed. The main goal of using hierarchical problem solving is to reduce the computational complexity. The applications of the theory to heuristic search, robot planning and function optimization are discussed.
Based on the theory, the combination of information from different levels of abstraction (granularities) is discussed. Some combination criteria are given. From one of these criteria we reveal the principle of the Dempster-Shafer combination rule in uncertain management. The same theory can be used to analyze the interdependency between quantitative, qualitative, fuzzy and other uncertain reasoning.